CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



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FIG. I. A: principle of measurement of flow and volume by constant-injection. Water-filled 

 beaker through which water flows at constant rate, F ml/min; into which dye is injected at constant 

 rate, / mg/min. Outflow of dye is F C, where C is concentration of dye, milligram per milliliter. 

 Eventually dye is distributed uniformly throughout beaker and when this occurs, despite continued 

 injection of dye, concentration throughout beaker is constant at Cmax. Whence inflow, /, = out- 

 flow, F Cmax- Dye in the beaker at the time C^ax is just reached is Cmax X V, volume of the sys- 

 tem between site of dye injection and outflow, and this amount of dye is difference between input 

 and output up to the time Cmax is reached. B: concentration of dye at outflow as a function of 

 time during constant-injection and subsequently. Shaded area, when multiplied by flow, F, is 

 quantity of dye remaining in system at concentration Cmax, and so measures volume. Plateau con- 

 centration is IIF. [From Zierler (38).] 



Stewart's method was not used widely until 

 Hamilton and his colleagues, after a lapse of several 

 decades, systematically improved the technique, 

 explored its possibilities, and popularized it (11, 12, 

 20). Hamilton used a dye which, on introduction 

 into the blood stream, bound to serum proteins, 

 chiefly albumin, and therefore passed out of the 

 vascular system through capillary walls only to a 

 negligible extent during the time over which blood 

 flow was measured. 



Over the years two forms of injection have been 

 used, both by Stewart and by Hamilton. In one, 

 indicator is injected very rapidly into the blood 

 stream, so that the distribution of injection with time 

 is simply a spike, under ideal circumstances. This 

 will be referred to as "sudden-injection." In the 

 other, indicator is injected continuously at constant 

 rate. This will be referred to as '"constant-injection." 

 There are, of course, many other possible variations 

 but only a few have been used and these to no great 

 extent. 



The principle by which sudden- and constant- 

 injections of indicator lead to a measure of flow is 

 simple. The argvmients developed in this section and 

 in the following section on measurement of volume 

 follow those offered by Meier & Zierler (19) and by 

 Zierler (38). A very similar approach is given by 

 van der Feer (37) and by Burger and colleagues (3). 



Consider first the case of constant-injection (fig. i). 



Inject indicator at constant rate, / mg per min, into 

 a system of fixed volume through which fluid flows 

 at constant but unknown rate, F ml per min. It is 

 obvious intuitively that after awhile the system will 

 hold all the indicator it is going to contain, if there 

 is no recirculation, and the rate at which indicator 

 leaves the system will exactly equal the rate at which 

 it is introduced into the system. The rate at which it 

 leaves the system is the product of the measurable 

 concentration of indicator at the outflow from the 

 system and the unknown flow of fluid, or 



/ = F X C„ 



(2) 



where C.nax is the concentration at outflow. It is 

 called Cniax because it is the maximum and constant 

 (because / and F are constant) concentration at 

 outflow. There will be from onset of injection an 

 initial transient during which, in sequence, no indi- 

 cator appears at outflow, the first measurable quan- 

 tity of indicator appears, the concentration of indi- 

 cator at outflow increases, in a way which we need 

 not yet specify, until indicator is distributed uniformly 

 throughout the system and its output equals its input. 

 Similarly, an intuitive argument for the case of 

 flow measurement by sudden-injection can be made 

 as follows (fig. 2). Indicator injected suddenly into a 

 fluid system appears at the outflow from the system 

 in a concentration which is some curvilinear function 

 of time, c(l). The rate at which indicator leaves the 



